Heavy-tailed fluctuations and power law statistics pervade physics, finance, and economics, yet their origin is often ascribed to systems poised near criticality. Here, we show that such behavior can emerge far from instability through a universal mechanism of non-normal eigenvector amplification in multidimensional Kesten processes ${\mathbf{x}}_{t+1}={\mathbf{A}}_t{\mathbf{x}}_t+{\mathit{\eta }}_t$, where ${\mathbf{A}}_t$ are random interaction $N\times N$ matrices and ${\mathit{\eta }}_t$ represents external inputs, capturing the evolving interdependence among $N$ coupled components. Even when each random multiplicative matrix is spectrally stable, nonorthogonal eigenvectors generate transient growth that renormalizes the Lyapunov exponent and decreases the tail exponent, producing stationary power laws without eigenvalues crossing the stability boundary. We derive explicit relations, first in the form of bounds and subsequently as approximations, linking the Lyapunov exponent $\gamma \sim {\gamma }_0+ln\kappa$ and the tail index $\alpha \sim -2\gamma /{\sigma }_{\kappa }^2$ to the statistics of the condition number with mean $\kappa$ and standard deviation ${\sigma }_{\kappa }$, confirmed by numerical simulations. This framework offers a unifying geometric perspective that helps interpret diverse phenomena, including polymer stretching in turbulence, magnetic-field amplification in dynamos, volatility clustering, and wealth inequality in financial systems. Non-normal interactions provide a collective route to scale-free behavior in globally stable systems, defining a universality class where multiplicative feedback and transient amplification generate critical-like statistics without spectral criticality.

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• Critical phenomena
• Econophysics
• Mathematical physics
• Nonequilibrium statistical mechanics
• Phase transitions